Computer Science > Discrete Mathematics

Title:
A Tighter Insertion-based Approximation of the Crossing Number

Abstract: Let G be a planar graph and F a set of additional edges not yet in G. The
multiple edge insertion problem (MEI) asks for a drawing of G+F with the
minimum number of pairwise edge crossings, such that the subdrawing of G is
plane. Finding an exact solution to MEI is NP-hard for general F. We present
the first polynomial time algorithm for MEI that achieves an additive
approximation guarantee -- depending only on the size of F and the maximum
degree of G, in the case of connected G. Our algorithm seems to be the first
directly implementable one in that realm, too, next to the single edge
insertion. It is also known that an (even approximate) solution to the MEI
problem would approximate the crossing number of the F-almost-planar graph G+F,
while computing the crossing number of G+F exactly is NP-hard already when
|F|=1. Hence our algorithm induces new, improved approximation bounds for the
crossing number problem of F-almost-planar graphs, achieving constant-factor
approximation for the large class of such graphs of bounded degrees and bounded
size of F.